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A radioactive nucleus A with a half life T, decays into a nucleus B. At t = 0, there is no nucleus B. At sometime, the ratio of the number of B to that of A is 0.3. Then, t is given by -

1. t = T log 2 / log 1.3
2. t = T log 1.3 / log 2
3. t = T log (1.3)
4. t = T / log (1.3)

Correct answer: t = T log 1.3 / log 2

Solution

The correct option is derived from the relationship between the decay of nucleus A and the formation of nucleus B, where the ratio of B to A can be expressed in terms of the half-life and logarithmic functions. By applying the decay formula and rearranging it to solve for time t, we find that t is proportional to the logarithm of the ratio of the quantities, specifically log(1.3) over log(2), which reflects the exponential nature of radioactive decay.

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